Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Stability

Stability

The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...

Root Loci for Positive-Feedback Systems

Root Loci for Positive-Feedback Systems

The Hartley oscillator is a positive feedback system that sustains oscillations by feeding the output back to the input in phase, thereby reinforcing the signal. Positive feedback systems can be viewed as negative feedback systems with inverted feedback signals. In these systems, the root locus encompasses all points on the s-plane where the angle of the system transfer function equals 360 degrees.
The construction rules for the root locus in positive feedback systems are similar to those in...

Control System Problem

Control System Problem

In an open-loop system, such as a basic thermostat, the poles of the transfer function influence the system's response but do not determine its stability. However, when feedback is introduced to form a closed-loop system, such as an advanced thermostat that adjusts heating based on room temperature, stability is governed by the new poles of the closed-loop transfer function.
When forming a closed-loop system, issues can arise if the poles cross into the unstable region, leading to potential...

Time-Domain Interpretation of PD Control

Time-Domain Interpretation of PD Control

Proportional-Derivative (PD) control is a widely used control method in various engineering systems to enhance stability and performance. In a system with only proportional control, common issues include high maximum overshoot and oscillation, observed in both the error signal and its rate of change. This behavior can be divided into three distinct phases: initial overshoot, subsequent undershoot, and gradual stabilization.
Consider the example of control of motor torque. Initially, a positive...

Pole and System Stability

Pole and System Stability

The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's response.

Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

A Continuous Time Dynamical Turing Machine.

IEEE transactions on neural networks and learning systems·2024

Same author

Stability of heteroclinic cycles in ring graphs.

Chaos (Woodbury, N.Y.)·2022

Same author

Where Computation and Dynamics Meet: Heteroclinic Network-Based Controllers in Evolutionary Robotics.

IEEE transactions on neural networks and learning systems·2019

Same author

Noise-induced tipping under periodic forcing: Preferred tipping phase in a non-adiabatic forcing regime.

Chaos (Woodbury, N.Y.)·2019

Same author

A topographic mechanism for arcing of dryland vegetation bands.

Journal of the Royal Society, Interface·2018

Same author

Simple Rules Govern the Patterns of Arctic Sea Ice Melt Ponds.

Physical review letters·2018

Search research articles

Related Experiment Video

Updated: Jul 7, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control.

Claire M Postlethwaite¹, Mary Silber

¹Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA. c-postlethwaite@northwestern.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|February 1, 2008

Summary

This summary is machine-generated.

This study demonstrates that Pyragas control can stabilize unstable periodic orbits in complex systems like the Lorenz equations. This challenges previous beliefs and offers a new strategy for controlling bifurcations.

More Related Videos

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

Related Experiment Videos

Last Updated: Jul 7, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

Area of Science:

Nonlinear Dynamics
Control Theory
Bifurcation Theory

Background:

Unstable periodic orbits were previously thought difficult to stabilize with Pyragas time-delayed feedback control.
A recent study by Fiedler et al. proposed a counterexample using Hopf bifurcation normal forms.

Purpose of the Study:

To demonstrate that the stabilization mechanism identified by Fiedler et al. can be applied to higher-dimensional systems.
To show that Pyragas control can stabilize unstable periodic orbits arising from subcritical Hopf bifurcations in the Lorenz equations.

Main Methods:

Analysis of a specific codimension-two bifurcation relevant to the Hopf normal form.
Application of Pyragas-type time-delayed feedback control to the Lorenz equations.
Investigation of feedback gain matrix selection informed by prior theoretical examples.

Main Results:

The stabilization mechanism for Hopf normal forms was successfully applied to the Lorenz equations.
A relevant codimension-two bifurcation was identified in the Lorenz equations under Pyragas control.
The proposed control strategy proved effective over a wide range of parameters.

Conclusions:

Pyragas control can stabilize unstable periodic orbits originating from subcritical Hopf bifurcations in complex systems.
The findings suggest a practical strategy for selecting feedback gain matrices in Pyragas control.
This work extends the applicability of stabilization mechanisms beyond simplified models.